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Properties

Label 2-74-37.27-c3-0-3
Degree $2$
Conductor $74$
Sign $0.798 + 0.602i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−0.261 − 0.452i)3-s + (1.99 + 3.46i)4-s + (2.77 − 1.60i)5-s + 1.04i·6-s + (8.06 + 13.9i)7-s − 7.99i·8-s + (13.3 − 23.1i)9-s − 6.40·10-s + 4.89·11-s + (1.04 − 1.81i)12-s + (71.2 − 41.1i)13-s − 32.2i·14-s + (−1.44 − 0.836i)15-s + (−8 + 13.8i)16-s + (−16.5 − 9.56i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0502 − 0.0870i)3-s + (0.249 + 0.433i)4-s + (0.247 − 0.143i)5-s + 0.0711i·6-s + (0.435 + 0.753i)7-s − 0.353i·8-s + (0.494 − 0.857i)9-s − 0.202·10-s + 0.134·11-s + (0.0251 − 0.0435i)12-s + (1.51 − 0.877i)13-s − 0.615i·14-s + (−0.0249 − 0.0143i)15-s + (−0.125 + 0.216i)16-s + (−0.236 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.798 + 0.602i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.798 + 0.602i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.19124 - 0.399148i\)
\(L(\frac12)\) \(\approx\) \(1.19124 - 0.399148i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
37 \( 1 + (221. - 38.3i)T \)
good3 \( 1 + (0.261 + 0.452i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-2.77 + 1.60i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-8.06 - 13.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 4.89T + 1.33e3T^{2} \)
13 \( 1 + (-71.2 + 41.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (16.5 + 9.56i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-99.0 + 57.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 27.6iT - 1.21e4T^{2} \)
29 \( 1 - 100. iT - 2.43e4T^{2} \)
31 \( 1 - 229. iT - 2.97e4T^{2} \)
41 \( 1 + (209. + 362. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 241. iT - 7.95e4T^{2} \)
47 \( 1 - 314.T + 1.03e5T^{2} \)
53 \( 1 + (-2.68 + 4.65i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (441. + 255. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-111. + 64.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-188. - 327. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (182. + 315. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 70.8T + 3.89e5T^{2} \)
79 \( 1 + (294. - 170. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (561. - 973. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (867. + 500. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 483. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.80152651423057869322266858201, −12.66435835498133927154332819036, −11.70491245133004480275075440893, −10.62426961013409421394784450406, −9.283690185087571512256862059769, −8.507824268571637421713061226947, −6.96337151170224169660199304928, −5.48369789285967062321031732620, −3.37073896706378868671092251157, −1.28842771989462858954683984972, 1.56151947570836324572018052965, 4.18798512035705343873677262562, 5.92000329697329237071234108223, 7.28173982761336328270570326443, 8.321742640091239192051707066441, 9.728239307919077383314443630905, 10.70053380131299058269065849442, 11.64169143346528303159209813772, 13.56700349532113358976622729244, 13.98231517517571459088418844374

Graph of the $Z$-function along the critical line